Simplify and expand the following expression: $ \dfrac{5}{t - 7}- \dfrac{3}{5t + 10}- \dfrac{4}{t^2 - 5t - 14} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{3}{5t + 10} = \dfrac{3}{5(t + 2)}$ We can factor the quadratic in the third term: $ \dfrac{4}{t^2 - 5t - 14} = \dfrac{4}{(t - 7)(t + 2)}$ Now we have: $ \dfrac{5}{t - 7}- \dfrac{3}{5(t + 2)}- \dfrac{4}{(t - 7)(t + 2)} $ The least common multiple of the denominators is: $ (t - 7)(t + 2)$ In order to get the first term over $(t - 7)(t + 2)$ , multiply by $\dfrac{5(t + 2)}{5(t + 2)}$ $ \dfrac{5}{t - 7} \times \dfrac{5(t + 2)}{5(t + 2)} = \dfrac{25(t + 2)}{(t - 7)(t + 2)} $ In order to get the second term over $(t - 7)(t + 2)$ , multiply by $\dfrac{t - 7}{t - 7}$ $ \dfrac{3}{5(t + 2)} \times \dfrac{t - 7}{t - 7} = \dfrac{3(t - 7)}{(t - 7)(t + 2)} $ In order to get the third term over $(t - 7)(t + 2)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{4}{(t - 7)(t + 2)} \times \dfrac{5}{5} = \dfrac{20}{(t - 7)(t + 2)} $ Now we have: $ \dfrac{25(t + 2)}{(t - 7)(t + 2)} - \dfrac{3(t - 7)}{(t - 7)(t + 2)} - \dfrac{20}{(t - 7)(t + 2)} $ $ = \dfrac{ 25(t + 2) - 3(t - 7) - 20} {(t - 7)(t + 2)} $ Expand: $ = \dfrac{25t + 50 - 3t + 21 - 20}{5t^2 - 25t - 70} $ $ = \dfrac{22t + 51}{5t^2 - 25t - 70}$